Related papers: Precovering and preenveloping ideals
Answering questions raised in \cite{Leonetti, Uzcategui} we characterize ideals $\mathcal I\subseteq \mathcal P(\omega)$ such that $c_{0,\mathcal I}$ is complemented in $\ell_\infty$ as exactly those ideals for which the space $K_{\mathcal…
Given a unital AF-algebra $A$ equipped with a faithful tracial state, we equip each (norm-closed two-sided) ideal of $A$ with a metrized quantum vector bundle structure, when canonically viewed as a module over $A$, in the sense of…
We study the category $\mathrm{Rep}(Q,\mathcal{M})$ of representations of a quiver $Q$ with values in an abelian category $\mathcal{M}$. Under certain assumptions, we show that every cotorsion pair $(\mathcal{A},\mathcal{B})$ in…
We prove the $K(\pi,1)$ conjecture for affine Artin groups: the complexified complement of an affine reflection arrangement is a classifying space. This is a long-standing problem, due to Arnol'd, Pham, and Thom. Our proof is based on…
We prove that, if $\textrm{GProj}$ is the class of all Gorenstein projective modules over a ring $R$, then $\mathfrak{GP}=(\textrm{GProj},\textrm{GProj}^\perp)$ is a cotorsion pair. Moreover, $\mathfrak{GP}$ is complete when all projective…
Given a complete hereditary cotorsion pair (A,B) in ModR, we construct a complete hereditary cotorsion pair in the derived category D(R) of unbounded complexes with respect to the proper class {\xi} of cohomologically ghost triangles…
A model structure on a category is a formal way of introducing a homotopy theory on that category, and if the model structure is abelian and hereditary, its homotopy category is known to be triangulated. So a good way to both build and…
Solid abelian groups, as introduced by Dustin Clausen and Peter Scholze, form a subcategory of all condensed abelian groups satisfying some ''completeness'' conditions and having favourable categorical properties. Given a profinite ring…
We describe a general correspondence between injective (resp. projective) recollements of triangulated categories and injective (resp. projective) cotorsion pairs. This provides a model category description of these recollement situations.…
Given a partial action of a discrete group $G$ on a Hausdorff, locally compact, totally disconnected topological space $X$, we consider the correponding partial action of $G$ on the algebra $L_c(X)$ consisting of all locally constant,…
We exhibit cocycles representing certain classes in the rational cohomology of of the general linear group with coefficients in the divided powers of a Frobenius twist of the adjoint representation. These classes' existence was anticipated…
This paper deals with the notion of grade of ideals with respect to torsion theories defined via some homological tools such as Ext-modules, Koszul cohomology modules, \v{C}ech and local cohomology modules over commutative rings which are…
In an abelian category $\mathscr{A}$, we can generate torsion pairs from tilting objects of projective dimension $\leq 1$. However, when we look at tilting objects of projective dimension $2$, there is no longer a natural choice of an…
A locally compact contraction group is a pair (G,f) where G is a locally compact group and f an automorphism of G which is contractive in the sense that the forward orbit under f of each g in G converges to the neutral element e, as n tends…
Given a complete hereditary cotorsion pair $(\mathcal{A},\mathcal{B})$ in a Grothendieck category $\mathcal{G}$, the derived category $\mathcal{D}(\mathcal{B})$ of the exact category $\mathcal{B}$ is defined as the quotient of the category…
We study the behaviour of modules $M$ that fit into a short exact sequence $0\to M\to C\to M\to 0$, where $C$ belongs to a class of modules $\mathcal C$, the so-called $\mathcal C$-periodic modules. We find a rather general framework to…
For any integer $n\ge 0$ and any ring $R$, \ $(\mathcal {PGF}_n, \ \mathcal P_n^\perp \cap \mathcal {PGF}^{\perp})$ proves to be a complete hereditary cotorsion pair in $R$-Mod, where $\mathcal {PGF}$ is the class of PGF modules, introduced…
We introduce the notion of homological systems $\Theta$ for triangulated categories. Homological systems generalize, on one hand, the notion of stratifying systems in module categories, and on the other hand, the notion of exceptional…
In 1960 Fuchs posed the problem of characterizing the groups which are the groups of units of commutative rings. In the following years, some partial answers have been given to this question in particular cases. In this paper we address…
This paper is to study cotorsion pairs and abelian model structures on some Morita rings \ $\Lambda =\left(\begin{smallmatrix} A & {}_AN_B {}_BM_A & B\end{smallmatrix}\right)$. From cotorsion pairs $(\mathcal U, \mathcal X)$ and $(\mathcal…